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On Solving Singular Diffusion Equations With Monte Carlo Methods
KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.ORCID iD: 0000-0002-7142-7103
2010 (English)In: IEEE Transactions on Plasma Science, ISSN 0093-3813, E-ISSN 1939-9375, Vol. 38, no 9, 2185-2189 p.Article in journal (Refereed) Published
Abstract [en]

Diffusion equations in one, two, or three dimensions with inhomogeneous diffusion coefficients are usually solved with finite-difference or finite-element methods. For higher dimensional problems, Monte Carlo solutions to the corresponding stochastic differential equations can be more effective. The inhomogeneities of the diffusion constants restrict the time steps. When the coefficient in front of the highest derivative of the corresponding differential equation goes to zero, the equation is said to be singular. For a 1-D stochastic differential equation, this corresponds to the diffusion coefficient that goes to zero, making the coefficient strongly inhomogeneous, which, however, is a natural condition when the process is limited to a region in phase space. The standard methods to solve stochastic differential equations near the boundaries are to reduce the time step and to use reflection. The strong inhomogeneity at the boundary will strongly limit the time steps. To allow for longer time steps for Monte Carlo codes, higher order methods have been developed with better convergence in phase space. The aim of our investigation is to find operators producing converged results for large time steps for higher dimensional problems. Here, we compare new and standard algorithms with known steady-state solutions.

Place, publisher, year, edition, pages
2010. Vol. 38, no 9, 2185-2189 p.
Keyword [en]
Diffusion equations, Monte Carlo methods, simulation, stochastic differential equations
National Category
Physical Sciences
URN: urn:nbn:se:kth:diva-27684DOI: 10.1109/TPS.2010.2057259ISI: 000283252500014ScopusID: 2-s2.0-77956617100OAI: diva2:380424
QC 20101221Available from: 2010-12-21 Created: 2010-12-20 Last updated: 2013-04-15Bibliographically approved
In thesis
1. On Monte Carlo Operators for Studying Collisional Relaxation in Toroidal Plasmas
Open this publication in new window or tab >>On Monte Carlo Operators for Studying Collisional Relaxation in Toroidal Plasmas
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns modelling of Coulomb collisions in toroidal plasma with Monte Carlo operators, which is important for many applications such as heating, current drive and collisional transport in fusion plasmas. Collisions relax the distribution functions towards local isotropic ones and transfer power to the background species when they are perturbed e.g. by wave-particle interactions or injected beams. The evolution of the distribution function in phase space, due to the Coulomb scattering on background ions and electrons and the interaction with RF waves, can be obtained by solving a Fokker-Planck equation.The coupling between spatial and velocity coordinates in toroidal plasmas correlates the spatial diffusion with the pitch angle scattering by Coulomb collisions.

In many applications the diffusion coefficients go to zero at the boundaries or in a part of the domain, which makes the SDE singular. To solve such SDEs or equivalent diffusion equations with Monte Carlo methods, we have proposed a new method, the hybrid method, as well as an adaptive method, which selects locally the faster method from the drift and diffusion coefficients. The proposed methods significantly reduce the computational efforts and improves the convergence.

The radial diffusion changes rapidly when crossing the trapped-passing boundary creating a boundary layer. To solve this problem two methods are proposed. The first one is to use a non-standard drift term in the Monte Carlo equation. The second is to symmetrize the flux across the trapped passing boundary. Because of the coupling between the spatial and velocity coordinates drift terms associated with radial gradients in density, temperature and fraction of the trapped particles appear. In addition an extra drift term has been included to relax the density profile to a prescribed one.

A simplified RF-operator in combination with the collision operator has been used to study the relaxation of a heated distribution function. Due to RF-heating the density of thermal ions is reduced by the formation of a high-energy tail in the distribution function. The Coulomb collisions tries to restore the density profile and thus generates an inward diffusion of thermal ions that results in a peaking of the total density profile of resonant ions.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. xi, 65 p.
Trita-EES, 2013:014
Fusion plasma, thermonuclear fusion, tokamak, Coulomb collisions, stochastic differential equations with singular diffusion coefficients, Monte Carlo schemes, spatial diffusion, modelling, Fokker-Planck equation, RF-heating.
National Category
Fusion, Plasma and Space Physics
urn:nbn:se:kth:diva-120590 (URN)978-91-7501-709-9 (ISBN)
Public defence
2013-05-13, F3, Lindstedtsvägen 26, KTH, Stockholm, 14:00 (English)

QC 20130415

Available from: 2013-04-15 Created: 2013-04-12 Last updated: 2013-10-18Bibliographically approved

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